The windswept Helsinki Cathedral, a Lutheran church modeled after the (Orthodox) Saint Isaac’s Cathedral in St. Petersburg. Yes, the sky is that blue.

]]>The *Open Logic Project* is live! From the website:

The

Open Logic Textis an open textbook on mathematical logic aimed at a non-mathematical audience, intended for advanced logic courses as taught in many philosophy departments. It is open-source: you can download the LaTeX code. It is open: you’re free to change it whichever way you like, and share your changes. It is collaborative: a team of people is working on it, using the GitHub platform, and we welcome contributions and feedback. And it is written with configurability in mind.

The Project was instigated and shepherded by Richard Zach, with contributions by Andy Arana, Jeremy Avigad, Gillian Russell, Nicole Wyatt and Audrey Yap (besides yours truly). Go and check it out!

]]>Groethendiek had an almost obsessive compulsion to generalize, abstracting from the clutter until just the conceptual scaffolding was left in view. Groethendiek’s student, Pierre Deligne (who himself won the Fields Medal in 1978 and the Abel Prize in 2013) is quoted in the obituary by *Le Monde*:

]]>He was unique in his way of thinking. He had to understand things from the broadest possible point of view, and once things are so conceived, the landscape becomes so clear that proofs appear to be almost trivial.

A little logic is a dangerous thing

Drink deep, or taste not the Fregean spring.

For a political philosopher to be in love with anyone is a sad thing.

Wonder what the other end of the conversation was like.

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For those who were wondering why there hadn’t been any new posts in a while: this beauty from the shores of Lake Como, that’s why.

]]>- A formula is satisfied by an assignment in if and only if the formula is satisfied, in , by every assignment that that differs from at most in the value .

The detour through satisfaction, while technically interesting, it is just that — a detour. It can also sometimes get pedagogically problematic, as students usually grasp the importance of truth while not always appreciating the necessity of satisfaction.

There is an alternative approach, which does away with the notion of satisfaction, but uses a variant notion of a model. According to this approach, we only allow models that are “rich” in that every object is the denotation of a constant in the language. The truth clause for the universal quantifier is then as follows:

- A sentence is true in if and only if is true in for each constant .

(I believe this definition is fairly standard in some textbooks, but I can’t track down a definition now, any references appreciated). While this definition sidesteps the notion of satisfaction, there are some drawbacks:

- The definition only applies to “rich” models, while models in which some elements are not denoted by constants are also very natural. Moreover, we are forced to consider truth of sentence in a model having a different signature than itself.
- The language changes with the model: for each model , the definition applies only to the expanded language introducing new constants for each member of .
- The definition forces us to consider uncountable languages, in order to assess truth in uncountable models.
- The expanded signature affects the available automorphism of a model. For example, the model has infinitely many automorphisms in the signature with the relation only, but only one if we add the constant .

So here is a way of combining these two approaches in such a way that:

- There is no detour through satisfaction.
- The language is countable and independent of the model in which a sentence is evaluated for truth (although the language will be expanded, just not in a way determined by the model).

Fix a basic language , and let be a countable set of new individual constants. For each let (so that ). We define the notion of truth by induction on -sentences, simultaneously for every . The quantifier case:

- An -sentence is true in the -structure if and only if the -sentence is true in every -expansion of (i.e., if is true in every -structure that differs from only in that it assigns a denotation to ).

So, when is an -sentence, the definition gives us a direct account of the truth or falsity of is a model having the same signature.

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